# Questions tagged [integers]

Questions about properties of, working with and algorithms on integers.

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### Linear time alg for checking if numbers in a list are close

Suppose we have an unsorted list of $n$ integers. What is the best algorithm we could use to detect if there exists a pair of ints that are 'close' to each other. Close being within some arbitrary int ...
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### Probability of overflow in a summation of fixed-size signed integers

How can I estimate the probability that the sum $S_n$ of $n$ uniform random 48-bit signed integers overflows a 64-bit signed integer? Edit: the overflow can occur at any step, not only on the final ...
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1 vote
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### Fast string-to-integer algorithm

I work with big integers that I need to convert to and from strings, and to and from different bases. In JavaScript, BigInt allows to convert an integer to a string in a chosen base by doing: ...
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### better ways to integer interpolation?

I made some code about "integer interpolation" for running approximate alpha blending at FPGA which have low quantities of logic gate. Let's refer to "II" as integer interpolation. ...
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### What is the purpose of padding bits in primitive integer types?

The C standard carves out room for implementations which have padding bits in their integer types. Padding bits affect the size of an integer but not the number of possible values they can contain. I ...
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### Optimal reassociations is NP-hard?

Consider signed integers with common addition and multiplication. Reassociation of expression is another equivalent form. Say expressions: ...
3k views

### Is there an efficient algorithm to find whether an integer is a prime power?

There's a sentence in the current version of the Wikipedia page for Shor's algorithm which states: we can use efficient classical algorithms to check if $N$ is a prime power. No reference is provided ...
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### Prove HAKMEM Item 23: connection between arithmetic operations and bitwise operations on integers

Prove that for $A, B \in \mathbb{Z}$, $A + B = (A \operatorname{\&} B) + (A \mid B) = (A \oplus B) + 2(A \operatorname{\&} B)$ where $\&$ is bitwise AND, $|$ is bitwise OR and $\oplus$ is ...
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1 vote
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### Complexity of multiplication when computing product of $m$ integers

I designed an algorithm where, at some point, I need to compute the product of a list of integers $n_1,\dots,n_m$ (possibly, there are repetitions in the list). The integers themselves do not depend ...
106 views

### Using Hashing to count the number of occurrences of a pattern within an integer array

So I have a problem that is ,I have an integer array and first I define an interval as a good interval iff, within the interval every integer appears an even (including zero) number of times. I want ...
86 views

### How to check if an positive integer can be represented as a sum of integers

How to determine if an integer x > 0 can be represented as 20n + 50m + 100k, where n,m,k >= 0 and are integers.
1 vote
1k views

### time complexity to convert string to integer and vice a versa

just given solution on this post i had mention the time complexity to convert string to int is O(n), also verifying this post now in one of comment fellow SO user, corrected me with example also ...
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### Efficient algorithm to "lift" a number in CRT representation mod r to mod $r^2$

Integers between 0 and a square-free number $r$ minus one can be represented by their value modulo each of $r$'s prime factors, according the Chinese remainder theorem. Given a number represented like ...
63 views

### Java | If a=250, b=3, c=(a + b/2)/b * b, Why doesn't c equal 251 but rather 249?

If int a = 250; int b = 3; int c = (a + b/2)/b * b; Why doesn't c equal 251, but rather 249? How is it that ...
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### Approximate x*(a/b)^(c/d) using integer arithmetic only (assembler)

0 < x,a,b,c,d < M are all positive integers (uint64). also, a<b if that helps. we have assembler (integer only) operations available (e.g. division only yields integers). we want to ...
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### $y$-Fast Tries: Why not partition into groups of $\Theta(\log^2 M)$ elements?

The $y$-fast trie is a data structure for storing a sorted collection of $n$ integers from the range $[0, M)$. It builds on the $x$-fast trie, which also stores elements in this range. The space usage ...
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### Batch rounding with preservation of a sum

I have a sequence of floating point numbers. I want to map each of them to one of their closest integers. There is one rule: Sum of integers must be as close to the sum of original numbers as possible ...
158 views

### Prove a greedy algorithm that obtains the minimum integer with at most k adjacent swaps is correct

This problem is from LeetCode. You're given a string num representing the digits of a very large integer and an integer k. You are allowed to swap any two adjacent digits of the integer at most k ...
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### Integer decomposition algorithm

Suppose I have a 32-bit integer $x$, I want to find $\{ x_i \}_{i \in 1\dots\ell}$ such that $x = e + \sum_{i=1}^\ell x_i \cdot 2^{32 - B\cdot i}$ where the error $e$ is as small as possible. The ...
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### What is the size of the largest numbers (integers) that current supercomputers can handle?

I am interested in designing number theory experiments and interested in knowing current capabilities of supercomputers to handle extremely large numbers. I'm interested in learning the size of the ...
1 vote
55 views

### Efficient comparison, using only sum, product, difference, and conditional jump if zero

I was wondering how small we could make the instruction set of a typical machine that supports a single datatype: arbitrary integers. If you need a heap, you declare an integer variable $h$ where you ...
1 vote
326 views

### Can Radix Sort be modified for signed ints and/or floats?

A few months ago I learned about the magic that allows radix sort to run in O(n) time and space. Most tutorials on radix sort say it is useful for very large ...
41 views

### Integer-only computation rounds down fractions in division require extra work?

In integer-only computation, a fraction like 5/2 is rounded down to 2. Is this any extra work at the ALU level, or, is the way it does division, at the level of logical gates, automatically outputting ...
1 vote
71 views

### What does it mean for an integer to belong to the halting problem?

I have come across the description of a function $F: \mathbb{N} \to \mathbb{N}$ where the function is defined one way for $n \in \mathcal{H}$ and another way for $n \notin \mathcal{H}.$ In this ...
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### How to represent $x$ in hexadecimal form where $x=2^n$?

When a value $x$ is a power of $2$, that is, $x = 2^n$ for some nonnegative integer n, we can readily write $x$ in hexadecimal form by remembering that the binary representation of $x$ is simply $1$ ...
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### Efficient Algorithm to Find the Closest Integer Representation, in the Form $A\times\frac{N}{D}$ for a Value

The Problem I am working on a problem that boils down to finding the closest representation of an arbitrary number ($x$) in the form: $$x = A\times\frac{N}{D}$$ Where $A$ is a 32-bit integer, and $N$ ...
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1 vote
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### How to rewrite a function such that integer division is applied before multiplication

Given the following function $$f(x,y) = (x \cdot y + 999)\; \text{div} \; 1000$$ where $x \in \{0, 1, 2, \dots, 2^{63}-1\}$, $y \in \{1, 2, 3, \dots, 500\}$, and the div operator is defined to round ...
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### Given $n$ sets of matrices, find $n$ matrices that have the least number of LCDs among their entries

Let's say I have $n$ sets of matrices  A = \left\{\begin{pmatrix} 2 & 4 & 17\\ 5 & 6 & 9\\ \end{pmatrix} \begin{pmatrix} 2 & 4 & 18\\ 5 & 6 & 9\\ \end{pmatrix} \right\...
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### while loop for bitwise conversion program

int counter(char c){ int count = 0; while (c!=0){ if((c & 1) !=0) count++; c = c >> 1; } } I am trying to understand a ...
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### For two large (10^6) integers A and B, find the number of set bits in the result of their multiplication

I've recently stumbled upon a problem that I struggle realy hard with ever since. There are two large decimal integers, order of magnitude $10^6$. The task is to find how many bits in the binary ...
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### Efficient triangular decomposition of an integer

Euclidean division is an iterative process that has been made super-efficient at the CPU level, right? Its specification is that if I do (q, r) = f(n, d), I get ...
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### What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String

What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String number representation while ...
206 views

### How would it be possible that primality testing is in P, but not factorization?

Suppose that P != NP. Then there exists 3SAT formulas such that their satisfiability is computationally "evil" (i.e, the satisfiability can be exponentially hard to determine in the size of ...
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### Do there exist fast multiplication algorithms for two integers with one of them being static?

Let N and M be arbitrary 1024+ bit integers. The objective is to compute the product of N and M (2048+ bits) There exist various multiplication algorithms for various bit lengths (ex library: GMP). ...
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### How computer understand to either represent the original bits or two's complement as value?

Important note I know that this question may seem too simple for you scientists; however, here is the best place that I know to post it. Question Suppose we have an ...
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### Maximizing integer sets intersection (with integer delta)

There are two sets of integers with different numbers of items in them. ...
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1 vote
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### Overlapping between two intervals: reasoning / algorithm to find the set of disjoint and overlapping intervals

Consider the positive integers {1, 2, 3, 4, ...} and the corresponding Integer Number Line. Suppose we have four integer numbers, A, B, C and D. For example: ...
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### pseudocode to split number value to array of numbers

How to write a pseudocode to split a number value to array of numbers? Let's say the number value is 12345. I need to convert it into [1,2,3,4,5].
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### What computational model supports arbitrarily sized integers?

I want to do some research, but I don't think it's important the number of bits it takes to represent the integer input and arithmetic on the abstract machine. So what is the model that addresses ...
360 views

### Algorithm for smallest number in array larger than threshold

We define the problem SmallestAbove as follows: Given an array $A$ of $n$ integers and a value $v$, compute the smallest value in $A$ that is strictly greater than $v$. Return $\infty$ if no such ...
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### Quick calculation for $x^y \bmod 2^d$

I need to calculate $x^y \bmod 2^d$ in $O(d)$ summations/bitwise operations and $1$ multiplication by $y$. $x$ is restricted to be odd, $d\geq 3$. $a$-bit arithmetic (for any $a$) is allowed, as this ...
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Suppose that sequence $A$ contains $n$ integers $a_1,a_2,a_3,\ldots,a_n$ and sequence $B$ contains $m$ integers $b_1,b_2,b_3,\ldots,b_m$. We know that $m \geq n$. We assume without loss of generality ...